Applying the
Power Rule
Master the most fundamental differentiation rule: learn how to find the derivative of any power of \(x\) quickly and confidently.
AP Calculus AB
What Is a Derivative?
A derivative measures the instantaneous rate of change of a function. If \( f(x) \) describes position, then \( f'(x) \) tells you the velocity: how fast the value is changing at every point.
Why the Power Rule Matters
Speed
Differentiate polynomials in seconds instead of computing limits term by term.
Foundation
The Power Rule is the building block for the Product Rule, Quotient Rule, and Chain Rule.
AP Exam
Nearly every free-response and multiple-choice question on derivatives uses the Power Rule.
Visualizing Slope on \( f(x) = x^2 \)
The Power Rule: Formal Statement
"Bring the exponent down as a coefficient, then reduce the exponent by one."
\( x^5 \)
\( \frac{d}{dx}[x^5] = 5x^4 \)
\( x^{-2} \)
\( \frac{d}{dx}[x^{-2}] = -2x^{-3} \)
\( x^{1/2} \)
\( \frac{d}{dx}[x^{1/2}] = \frac{1}{2}x^{-1/2} \)
Pairing with the Constant Multiple Rule
When a constant \( c \) multiplies a function, the constant passes through the derivative:
\( 7x^3 \)
\( \frac{d}{dx}[7x^3] = 7 \cdot 3x^2 = 21x^2 \)
\( -4x^{10} \)
\( \frac{d}{dx}[-4x^{10}] = -40x^9 \)
The Sum and Difference Rule
Derivatives distribute over addition and subtraction. This means you can differentiate term by term:
Power Rule vs. Exponential Rule
| Power Rule | Exponential Rule | |
|---|---|---|
| Form | \( x^n \) (variable base) | \( a^x \) (variable exponent) |
| Derivative | \( n \cdot x^{n-1} \) | \( a^x \cdot \ln a \) |
| Example | \( x^3 \to 3x^2 \) | \( 2^x \to 2^x \ln 2 \) |
| Key idea | Exponent comes down | Function stays, multiply by \( \ln a \) |
How to Apply the Power Rule
1
Rewrite every term as a power of \( x \). Convert roots to fractional exponents and fractions to negative exponents.
2
Identify the coefficient \( c \) and the exponent \( n \) for each term.
3
Apply the rule: multiply the coefficient by the exponent, then reduce the exponent by 1: \( c \cdot n \cdot x^{n-1} \).
4
Simplify the result. Combine constants, convert back to radical or fraction notation if required.
Example 1: Basic Polynomial
1
Differentiate each term: \( 4 \cdot 3x^{2} = \) \( 12x^2 \)
2
\( -6 \cdot 2x^{1} = \) \( -12x \)
3
\( 1 \cdot x^{0} = \) \( 1 \) (since \( x^0 = 1 \))
4
Derivative of the constant \( -9 \) is \( 0 \).
Example 2: Radicals and Negative Exponents
1
Rewrite: \( x^{1/2} + 3x^{-2} \)
2
Apply Power Rule: \( \frac{1}{2}x^{-1/2} + 3(-2)x^{-3} \)
3
Simplify: \( \frac{1}{2\sqrt{x}} - \frac{6}{x^3} \)
Example 3: Multi-Step Expression
1
Divide each term by \( x \): \( 2x^4 - 3x^{-2/3} \)
2
Differentiate: \( 2(4)x^3 - 3\!\left(-\frac{2}{3}\right)x^{-5/3} \)
3
Simplify: \( 8x^3 + 2x^{-5/3} = 8x^3 + \frac{2}{x^{5/3}} \)
Graphical Interpretation
Special Cases of the Power Rule
Constants
\( \frac{d}{dx}[c] = 0 \). A constant is \( c \cdot x^0 \), so \( 0 \cdot x^{-1} = 0 \).
Linear
\( \frac{d}{dx}[x] = 1 \). Since \( x = x^1 \), we get \( 1 \cdot x^0 = 1 \).
\( n = -1 \)
\( \frac{d}{dx}\!\left[\frac{1}{x}\right] = -x^{-2} = -\frac{1}{x^2} \).
Common Mistakes to Avoid
✕ Forgetting to Rewrite
Trying to differentiate \( \sqrt{x} \) or \( \frac{1}{x^3} \) without first converting to \( x^{1/2} \) or \( x^{-3} \).
✕ Confusing with Exponentials
Applying the Power Rule to \( 2^x \). Remember: the Power Rule is for \( x^n \), NOT \( n^x \).
✕ Dropping the Constant
Writing \( \frac{d}{dx}[5x^3] = 3x^2 \) instead of \( 15x^2 \). The coefficient multiplies the result.
Power Rule Summary
| Original \( f(x) \) | Rewritten | Derivative \( f'(x) \) |
|---|---|---|
| \( 6x^4 \) | \( 6x^4 \) | \( 24x^3 \) |
| \( \sqrt{x} \) | \( x^{1/2} \) | \( \frac{1}{2}x^{-1/2} \) |
| \( \dfrac{1}{x^3} \) | \( x^{-3} \) | \( -3x^{-4} \) |
| \( \sqrt[3]{x^2} \) | \( x^{2/3} \) | \( \frac{2}{3}x^{-1/3} \) |
| \( -8 \) | \( -8x^0 \) | \( 0 \) |
Question 1 of 5
What is \( \frac{d}{dx}[x^7] \)?
The Power Rule says bring the exponent 7 down as a coefficient, then subtract 1 from the exponent: \( 7 \cdot x^{7-1} = 7x^6 \).
Question 2 of 5
Find \( \frac{d}{dx}\!\left[3x^4 - 2x + 5\right] \).
Differentiate term by term: \( 3 \cdot 4x^3 = 12x^3 \), \( -2 \cdot 1x^0 = -2 \), and the constant \( 5 \) vanishes. Result: \( 12x^3 - 2 \).
Question 3 of 5
What is \( \frac{d}{dx}\!\left[\sqrt{x}\right] \)?
Rewrite \( \sqrt{x} = x^{1/2} \). Apply the Power Rule: \( \frac{1}{2}x^{-1/2} = \frac{1}{2} \cdot \frac{1}{\sqrt{x}} = \frac{1}{2\sqrt{x}} \).
Question 4 of 5
The Power Rule cannot be directly applied to which of the following?
\( 5^x \) is an exponential function (variable in the exponent), not a power function. Its derivative is \( 5^x \ln 5 \), which requires the Exponential Rule, not the Power Rule.
Question 5 of 5
Find \( \frac{d}{dx}\!\left[\dfrac{4}{x^2} - \sqrt[3]{x}\right] \).
Rewrite: \( 4x^{-2} - x^{1/3} \). Differentiate: \( 4(-2)x^{-3} - \frac{1}{3}x^{-2/3} = -8x^{-3} - \frac{1}{3}x^{-2/3} \). Converting back: \( -\frac{8}{x^3} - \frac{1}{3x^{2/3}} \).